where a similar strategy was used in [1]} and [2]} and we drop the arguments of \(\Xi ^{(q)}\)  and \(\Xi ^{(q+1)}\)  as long as no confusion arises.
\(&Err_{\hat{\Xi }^{(q+1)}}(\hat{\Xi }^{(q)}) - Err_{\hat{\Xi }^{(q+1)}}(\Xi ^{(q)}_*)\\& = E[\hat{\Xi }^{(q)}]^2 - E[\Xi ^{(q)}_*]^2 + 2E\bigg [ (\Xi ^{(q)}_* - \hat{\Xi }^{(q)})\max _{a^{(q+1)}} \Xi ^{(q+1)}_*\bigg ] \\& ~ ~ -2E\bigg [(\max _{a^{(q+1)}} \Xi ^{(q+1)}_* - \max _{a^{(q+1)}} \hat{\Xi }^{(q+1)}) (\Xi ^{(q)}_* - \hat{\Xi }^{(q)})\bigg ] \\& = E[\hat{\Xi }^{(q)}]^2 - E[\Xi ^{(q)}_*]^2 + 2E\bigg [ (\Xi ^{(q)}_* - \hat{\Xi }^{(q)})\Xi ^{(q)}_*\bigg ] \\& ~ ~ -2E\bigg [(\max _{a^{(q+1)}} \Xi ^{(q+1)}_* - \max _{a^{(q+1)}} \hat{\Xi }^{(q+1)}) (\Xi ^{(q)}_* - \hat{\Xi }^{(q)})\bigg ] \\& = E[\hat{\Xi }^{(q)}- \Xi ^{(q)}_*]^2 -2E\bigg [(\max _{a^{(q+1)}} \Xi ^{(q+1)}_* - \max _{a^{(q+1)}} \hat{\Xi }^{(q+1)}) (\Xi ^{(q)}_* - \hat{\Xi }^{(q)})\bigg ] \\& \ge E[\hat{\Xi }^{(q)}- \Xi ^{(q)}_*]^2 -2E^{1/2}\bigg [\max _{a^{(q+1)}} \Xi ^{(q+1)}_* - \max _{a^{(q+1)}} \hat{\Xi }^{(q+1)}\bigg ]^2E^{1/2}\bigg [\Xi ^{(q)}_* - \hat{\Xi }^{(q)}\bigg ]^2 \\& \ge E[\hat{\Xi }^{(q)}- \Xi ^{(q)}_*]^2 -2\sqrt{LE\bigg [\Xi ^{(q+1)}_* - \hat{\Xi }^{(q+1)}\bigg ]^2}E^{1/2}\bigg [\Xi ^{(q)}_* - \hat{\Xi }^{(q)}\bigg ]^2 \\& \ge E[\hat{\Xi }^{(q)}- \Xi ^{(q)}_*]^2 -\frac{1}{2} \lbrace 4LE\bigg [\Xi ^{(q+1)}_* - \hat{\Xi }^{(q+1)}\bigg ]^2+ E\bigg [\Xi ^{(q)}_* - \hat{\Xi }^{(q)}\bigg ]^2\rbrace \\&= E[\hat{\Xi }^{(q)}- \Xi ^{(q)}_*]^2 - 2L E[\hat{\Xi }^{(q+1)}- \Xi ^{(q+1)}_*]^2,\) 
